TOKYO, March 8 (Reuters) - The dollar recovered ground on Thursday, drawing relief from positive labour market data and the White House saying Canada and Mexico, and possibly other countries, may be exempted from planned U.S. import tariffs on steel and aluminium.

Elsewhere on the currency markets, the euro trod water ahead of a European Central Bank meeting later in the global day, that is expected to leave policy rates unchanged for now, but may provide clues to the future.

The dollar had weakened sharply following the resignation on Tuesday of Gary Cohn, the top economic advisor to the White House who was seen as seen as a bulwark against protectionism in the Trump administration.

His departure had fanned fears of a potential global trade war if U.S. President Donald Trump pressed ahead with proposals for tariffs on all imports of steel and aluminium.

Broader financial markets grew calmer, with Wall Street paring losses overnight, after the White House raised the possibility of exemptions. Some dealers have bet on the possibility that the tariff threat was a negotiating ploy in trade talks with neighbours.

Amid growing pressure to exempt U.S. allies, Trump is expected to sign a proclamation establishing the steel and aluminium tariffs during a ceremony scheduled for 2030 GMT.

The dollar’s bounce was stoked further by Wednesday’s data on domestic private hiring and labour costs that reinforced the view of underlying strength in the U.S. economy.

The U.S. currency was steady at 106.130 yen after slipping to as low as 105.450 the previous day in reaction to Cohn’s departure.

The dollar index against a basket of six major currencies was effectively flat at 89.592 after pulling away from a two-week trough of 89.407 set the previous day.

“The market has managed to digest Cohn’s resignation,” said Junichi Ishikawa, senior FX strategist at IG Securities in Tokyo. “That said, ‘Trump risk,’ which are developments associated with the U.S. president and his circle, will continue to impact the market as once of the main themes of 2018,”

The euro was little changed at $1.2409 after retracing a bounce earlier on Wednesday to a 2-1/2-week peak of $1.2447.

Focus for the common currency was on the ECB’s policy decision at 1245 GMT. The central bank is all but certain to keep policy unchanged but may tweak its communication stance to offer at least a few clues about its progress towards ending its unprecedented bond purchases later this year.

“Under normal circumstances the ECB decision would not be a risk event as the central bank is expected to stand pat on policy. But the ECB’s stance will be watched carefully in the wake of political confusion in the United States,” Ishikawa at IG Securities said.

The Canadian dollar was at C$1.2914 per dollar, having weakened to C$1.3002 on Wednesday but recovering on the prospect that Trump’s tariffs may exempt Canada and Mexico.

The Mexican currency was at 18.72 pesos per dollar following its recovery from 18.90 touched on Wednesday.

The Australian and New Zealand dollars were little changed at $0.7829 and $0.7286, respectively. (Editing by Simon Cameron-Moore)